Category theory

branch of mathematics studying categories, functors, and natural transformations

In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the natural numbers with addition form a monoid, the identity element being .

abstract object in mathematics

algebraic structure of objects and morphisms between objects, which can be associatively composed if the (co)domains agree

collection of maps in which all map compositions starting from the same set and ending with the same set give the same result

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

central object of study in category theory

In mathematics, specifically category theory, a subcategory of a category \mathcal{C} is a category \mathcal{S} whose objects are objects in \mathcal{C} and whose morphisms are morphisms in \mathcal{C} with the same identities and composition of morphisms. Intuitively, a subcategory of \mathcal{C} is a category obtained from \mathcal{C} by "removing" some of its objects and arrows.

category equipped with a faithful functor to the category of sets

generalization of the kernel of a homomorphism in category theory

monoid in an endofunctor category

correspondence between properties of a category and its opposite

transitive set closed under the operations of parameterized union, power set and unordered pairs

The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the corank of .

abstract mathematics relationship

generalized directed graph which is permitted to have multiple arcs and loops

category constructed from another category C, whose objects are the same as those of C, whose morphisms from X to Y are the same as the morphisms in C from Y to X

object 𝑋 in an abelian category such that hom(–,𝑋) is an exact functor to the opposite category of the category of abelian groups

Category theory constructs

two kinds of possible inverses of a morphism in a category

endomorphism algebra of an abelian group

mathematical product of two categories, in category theory

In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an algebra over O to be a set together with concrete operations on this set that behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group

a set equipped with a choice of a specific element

In functional programming, the concept of catamorphism (from the Ancient Greek: "downwards" and "form, shape") denotes the unique homomorphism from an initial algebra into some other algebra.

thumb|The commutative diagram, which defines a property required by morphisms of the original Category (mathematics)|category, so that they can be morphisms of the newly defined category of F-algebras.

tongue-in-cheek description of category theory and abstract mathematics

category whose hom sets have additional structure

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.

mathematical category with weak equivalences, fibrations and cofibrations

In computer science, corecursion is a type of operation that is dual to (structural) recursion. Whereas recursion consumes a data structure by first handling the topmost layer before descending into its inner parts, corecursion produces a data structure by first defining the topmost layer before defining its inner parts. Corecursion is a particularly important in total languages, as it allows encoding potentially non-terminating computations in a context where every function must terminate. It is supported by theorem provers Agda and Rocq.

In algebra, the coimage of a homomorphism

generalization (and categorification) of a sheaf; a fibered category that admits effective descent

term in category theory

generalization of a fiber bundle dropping the condition of a local product structure

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups. Semigroupoids have applications in the structural theory of semigroups.

In computer science, coinduction is a technique for defining and proving properties of systems of concurrent interacting objects.

in category theory

research proposal by A. Grothendieck

book by Saunders Mac Lane

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory.

concept used in defining a Grothendieck topology; generalization of the collection of open subsets of an open set

In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F, with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy evaluation, infinite data structures, such as streams, and also transition systems.